# Content-Type: text/shitpost

Subject: Non-transitive relations
Date: 2018-05-02T18:12:47
Newsgroup: talk.mjd.math.non-transitive-relations
Message-ID: <64b0f1e9b2f83316@shitpost.plover.com>
Content-Type: text/shitpost

The phases of an eight-phase dual-ring intersection provide an interesting example of a non-transitive relation.

Let !!a!! and !!b!! be phases, and write !!a\sim b!! when the phases are compatible, which means that traffic making those movements will not collide. This relation is reflexive and symmetric, but not transitive, because for example we have !!\Phi6\sim \Phi1!! and !!\Phi1\sim \Phi5!! but not !!\Phi6\sim\Phi 5!!. A mathematician would have numbered the phases differently, but even with the standard numbering the rule is not too complicated:

$$a\sim b \text{ when any of these holds:} \begin{cases} a\in\{\Phi1, \Phi 2\}\text{ and }b\in\{\Phi5, \Phi6\},\text{ or} \\ a\in\{\Phi3, \Phi 4\}\text{ and }b\in\{\Phi7, \Phi8\},\text{ or} \\ a = b \end{cases}$$